\(\int \frac {e^{\arctan (a x)}}{(c+a^2 c x^2)^4} \, dx\) [251]

Optimal result

Integrand size = 19, antiderivative size = 116 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {144 e^{\arctan (a x)}}{629 a c^4}+\frac {e^{\arctan (a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac {30 e^{\arctan (a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac {72 e^{\arctan (a x)} (1+2 a x)}{629 a c^4 \left (1+a^2 x^2\right )} \]

[Out]

144/629*exp(arctan(a*x))/a/c^4+1/37*exp(arctan(a*x))*(6*a*x+1)/a/c^4/(a^2*x^2+1)^3+30/629*exp(arctan(a*x))*(4*
a*x+1)/a/c^4/(a^2*x^2+1)^2+72/629*exp(arctan(a*x))*(2*a*x+1)/a/c^4/(a^2*x^2+1)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5178, 5179} \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {72 (2 a x+1) e^{\arctan (a x)}}{629 a c^4 \left (a^2 x^2+1\right )}+\frac {30 (4 a x+1) e^{\arctan (a x)}}{629 a c^4 \left (a^2 x^2+1\right )^2}+\frac {(6 a x+1) e^{\arctan (a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}+\frac {144 e^{\arctan (a x)}}{629 a c^4} \]

[In]

Int[E^ArcTan[a*x]/(c + a^2*c*x^2)^4,x]

[Out]

(144*E^ArcTan[a*x])/(629*a*c^4) + (E^ArcTan[a*x]*(1 + 6*a*x))/(37*a*c^4*(1 + a^2*x^2)^3) + (30*E^ArcTan[a*x]*(
1 + 4*a*x))/(629*a*c^4*(1 + a^2*x^2)^2) + (72*E^ArcTan[a*x]*(1 + 2*a*x))/(629*a*c^4*(1 + a^2*x^2))

Rule 5178

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n - 2*a*(p + 1)*x)*(c + d*x^2)
^(p + 1)*(E^(n*ArcTan[a*x])/(a*c*(n^2 + 4*(p + 1)^2))), x] + Dist[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2))
), Int[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1]
&&  !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]

Rule 5179

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcTan[a*x])/(a*c*n), x] /; Fre
eQ[{a, c, d, n}, x] && EqQ[d, a^2*c]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{\arctan (a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac {30 \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{37 c} \\ & = \frac {e^{\arctan (a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac {30 e^{\arctan (a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac {360 \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{629 c^2} \\ & = \frac {e^{\arctan (a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac {30 e^{\arctan (a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac {72 e^{\arctan (a x)} (1+2 a x)}{629 a c^4 \left (1+a^2 x^2\right )}+\frac {144 \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx}{629 c^3} \\ & = \frac {144 e^{\arctan (a x)}}{629 a c^4}+\frac {e^{\arctan (a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac {30 e^{\arctan (a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac {72 e^{\arctan (a x)} (1+2 a x)}{629 a c^4 \left (1+a^2 x^2\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {17 c e^{\arctan (a x)} (1+6 a x)+6 \left (c+a^2 c x^2\right ) \left (5 e^{\arctan (a x)} (1+4 a x)+12 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} (-i+a x) (i+a x) \left (3+2 a x+2 a^2 x^2\right )\right )}{629 a c^2 \left (c+a^2 c x^2\right )^3} \]

[In]

Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^4,x]

[Out]

(17*c*E^ArcTan[a*x]*(1 + 6*a*x) + 6*(c + a^2*c*x^2)*(5*E^ArcTan[a*x]*(1 + 4*a*x) + (12*(1 - I*a*x)^(I/2)*(-I +
a*x)*(I + a*x)*(3 + 2*a*x + 2*a^2*x^2))/(1 + I*a*x)^(I/2)))/(629*a*c^2*(c + a^2*c*x^2)^3)

Maple [A] (verified)

Time = 32.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61

method	result	size
gosper	\(\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (144 a^{6} x^{6}+144 a^{5} x^{5}+504 a^{4} x^{4}+408 a^{3} x^{3}+606 a^{2} x^{2}+366 a x +263\right )}{629 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\)	\(71\)
parallelrisch	\(\frac {144 a^{6} {\mathrm e}^{\arctan \left (a x \right )} x^{6}+144 a^{5} {\mathrm e}^{\arctan \left (a x \right )} x^{5}+504 a^{4} {\mathrm e}^{\arctan \left (a x \right )} x^{4}+408 a^{3} x^{3} {\mathrm e}^{\arctan \left (a x \right )}+606 x^{2} {\mathrm e}^{\arctan \left (a x \right )} a^{2}+366 \,{\mathrm e}^{\arctan \left (a x \right )} a x +263 \,{\mathrm e}^{\arctan \left (a x \right )}}{629 c^{4} \left (a^{2} x^{2}+1\right )^{3} a}\)	\(102\)

[In]

int(exp(arctan(a*x))/(a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)

[Out]

1/629*exp(arctan(a*x))*(144*a^6*x^6+144*a^5*x^5+504*a^4*x^4+408*a^3*x^3+606*a^2*x^2+366*a*x+263)/(a^2*x^2+1)^3
/c^4/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {{\left (144 \, a^{6} x^{6} + 144 \, a^{5} x^{5} + 504 \, a^{4} x^{4} + 408 \, a^{3} x^{3} + 606 \, a^{2} x^{2} + 366 \, a x + 263\right )} e^{\left (\arctan \left (a x\right )\right )}}{629 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

1/629*(144*a^6*x^6 + 144*a^5*x^5 + 504*a^4*x^4 + 408*a^3*x^3 + 606*a^2*x^2 + 366*a*x + 263)*e^(arctan(a*x))/(a
^7*c^4*x^6 + 3*a^5*c^4*x^4 + 3*a^3*c^4*x^2 + a*c^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (107) = 214\).

Time = 7.51 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.43 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\begin {cases} \frac {144 a^{6} x^{6} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {144 a^{5} x^{5} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {504 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {408 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {606 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {366 a x e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {263 e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} & \text {for}\: a \neq 0 \\\frac {x}{c^{4}} & \text {otherwise} \end {cases} \]

[In]

integrate(exp(atan(a*x))/(a**2*c*x**2+c)**4,x)

[Out]

Piecewise((144*a**6*x**6*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*
a*c**4) + 144*a**5*x**5*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a
*c**4) + 504*a**4*x**4*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a*
c**4) + 408*a**3*x**3*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a*c
**4) + 606*a**2*x**2*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a*c*
*4) + 366*a*x*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a*c**4) + 2
63*exp(atan(a*x))/(629*a**7*c**4*x**6 + 1887*a**5*c**4*x**4 + 1887*a**3*c**4*x**2 + 629*a*c**4), Ne(a, 0)), (x
/c**4, True))

Maxima [F]

\[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

integrate(e^(arctan(a*x))/(a^2*c*x^2 + c)^4, x)

Giac [F]

\[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \]

[In]

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {144\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}}{629\,a\,c^4}+\frac {72\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{629\,a\,c^4\,\left (a^2\,x^2+1\right )}+\frac {30\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x+1\right )}{629\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (6\,a\,x+1\right )}{37\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \]

[In]

int(exp(atan(a*x))/(c + a^2*c*x^2)^4,x)

[Out]

(144*exp(atan(a*x)))/(629*a*c^4) + (72*exp(atan(a*x))*(2*a*x + 1))/(629*a*c^4*(a^2*x^2 + 1)) + (30*exp(atan(a*
x))*(4*a*x + 1))/(629*a*c^4*(a^2*x^2 + 1)^2) + (exp(atan(a*x))*(6*a*x + 1))/(37*a*c^4*(a^2*x^2 + 1)^3)